For puzzles, there is minimal value in preserving an incorrect answer except as a signpost to other would-be answerers that the answer you tried has been rejected.
If you have multiple possible answers, and think all of them have a reasonable likelihood of being correct, that might be a good reason to supply more than one answer to the same question. But if you're essentially replacing an existing but known-incorrect answer with a different attempt, there's often little reason to retain the original answer, so it is entirely reasonable to edit your former answer with your new attempt. This avoids cluttering up the answers section with answers known to be wrong (and also avoids drawing the downvotes that wrong answers often collect).
If you think the line of reasoning you took in a wrong answer is something others are very likely to also follow to the same result, it may be worth keeping that wrong answer around (and giving another attempt as a new answer) just so people can see that it's already been tried and found wanting. If the asker indicates your answer is wrong but you think there is substantial value in your answer (e.g. a sincere "this should be the answer" scenario), that might be another reason to keep an existing answer and add a new one for a subsequent attempt. Beyond these cases, I would not as a rule create a new answer for every attempt.
The system pops up a dialog box when you try to add a new answer to a question you already have an answer for. This is no accident. :)